Stanley M. Aronson: The redeeming virtues of chaos, and modern math

By Stanley M. Aronson

Monday

Dec 30, 2013 at 12:01 AM

In one of his many letters, Galileo Galilei (1564-1642) wrote: “To our natural and human reason, I say that these terms ‘large,’ ‘small,’ ‘immense,’ ‘minute,’ etc., are not absolute, but relative; the...

In one of his many letters, Galileo Galilei (1564-1642) wrote: “To our natural and human reason, I say that these terms ‘large,’ ‘small,’ ‘immense,’ ‘minute,’ etc., are not absolute, but relative; the same thing in comparison with various others may be called at one time ’immense’ and at another, ‘imperceptible.’ ”

Galileo’s statement — more a passing reflection than the setting forth of a firm principle — hints at his aversion to absolutist, unqualified declarations. He portrayed the cosmos as uninterrupted movement, an ever-evolving, ever-changing universe where there is no fixed point, no immoveable locus, where magnitudes can be judged solely by comparisons with like objects — in short, a world where words such as ‘most,’ ‘least’ and ‘furthest’ have become meaningless.

In such a world of untethered values and moral ambiguities, many perplexed persons feel a nostalgia, a longing for the reliable, for the unambiguous, the absolute. They seek more unconditional frameworks that offer them a comforting assurance; and they strive for an ethical foundation that does not shift from season to season. And so they express their support for absolute monarchies, religions without contingencies, absolute purities and of course, that wondrously illusion called the absolute truth. It is not for mere geologic reasons that the Rock of Gibraltar remains an enduring moral symbol.

The poignant desire for permanency, however, extends beyond the casual wishes expressed in daily conversation. Scientists, newly arrived to the domain of modern science, also find themselves reluctant to leave the warm and fuzzy comfort of classical Hellenic science.

The mathematics of Pythagoras and Euclid were emotionally and arithmetically reassuring. In clear, unambiguous terms, it set forth provable axioms about tangible structures such as rectangles, triangles and cones.

Numbers, of course, were crucial to classical mathematics, yet in their zeal for indestructible realities, early mathematicians did not recognize such arithmetical entities as zero, negative numbers or irrational numbers (those numbers that cannot be expressed as a simple ratio of numbers). Nor could irrational numbers be portrayed as finite decimals. Indeed, the German mathematician Leopold Kronecker (1823-1891) once declared: “God made the integers; the rest [irrational numbers] is the work of man.” The Hellenic world remained content with numbers that were tangible, positive and understandable. And to them, mathematics was a synonym for the truth, austere and cold perhaps, but still the perfect truth.

To the courageous mathematicians of the 17th century and beyond there was a limitless reality in the unreal world of irrational numbers, infinite numbers and transcendental numbers. A number called irrational was not insane but merely a number without a conventional, tidy ratio; and infinity was only the uncounted, not the uncountable.

Beyond the 17th century, the ground beneath the newer mathematics was never felt to be solid, never the same from day to day; and those mathematicians, still imprisoned in a three-dimensional world, felt unanchored.

Beyond the 17th Century mathematicians slowly accepted the notion that there were many systems, all equally true, constructed on different sets of axioms. Each system internally true — but incompatible with other systems. And therefore, to some, mathematics had no connection to pedestrian realities.

But mathematics never claimed to be a faithful representation of reality. Einstein (1879-1955) asked: “How can it be that mathematics, being after all a product of human thought (allegedly) independent of experience, is so admirably adapted to the objects of reality?” He then pointed out that conic sections in geometry also defined the orbits of planets, that the whorls of the chambered nautilus complied perfectly with the Fibonacci series of numbers, which also described the circular arrangement of sunflower seeds or the scales of the pine cone.

The world is surely complex, seemingly chaotic, and certainly made up of random events casting doubt on causality. And so modern mathematics has cautiously offered a theory of chaos, in the words of Edward Lorenz (1917-2008): “When the present determines the future, but the approximate present does not approximately determine the future.” Chaos, then, is a word that we have invented for something that just is not yet understood.